Symmetry protected topological orders of 1D spin systems with D2+T symmetry

TL;DR

This paper classifies all 16 symmetry protected topological phases in 1D integer spin chains with D2+T symmetry, characterizing them via boundary states, and demonstrates their realization in S=1 spin ladder models.

Contribution

It extends previous work by fully classifying 16 SPT phases under D2+T symmetry and linking them to measurable boundary properties.

Findings

All 16 SPT phases are characterized by boundary degeneracies.

Each phase can be experimentally distinguished by end spin measurements.

All phases are realizable in S=1 spin ladder models.

Abstract

In [Z.-X. Liu, M. Liu, X.-G. Wen, arXiv:1101.5680], we studied 8 gapped symmetric quantum phases in S=1 spin chains %/ladders which respect a discrete spin rotation $D_2 \subset SO(3)$ and time reversal $T$ symmetries. In this paper, using a generalized approach, we study all the 16 possible gapped symmetric quantum phases of 1D integer spin systems with only $D_2+T$ symmetry. Those phases are beyond Landau symmetry breaking theory and cannot be characterized by local order parameters, since they do not break any symmetry. They correspond to 16 symmetry protected topological (SPT) orders. We show that all the 16 SPT orders can be fully characterized by the physical properties of the symmetry protected degenerate boundary states (end `spins') at the ends of a chain segment. So we can measure and distinguish all the 16 SPT orders experimentally. We also show that all these SPT orders can be realized in S=1 spin ladder models. The gapped symmetric phases protected by subgroups of $D_2+T$ are also studied. Again, all these phases can be distinguished by physically measuring their end `spins'.